Matlab: Help neede to initialize matrices - matlab

Question

I am trying to implement the basic Equations for Kalman filter for the following 1 dimensional AR model:
x(t) = a_1x(t-1) + a_2x(t-2) + w(t)
y(t) = Cx(t) + v(t);
The KF state space model :
x(t+1) = Ax(t) + w(t)
y(t) = Cx(t) + v(t)
w(t) = N(0,Q)
v(t) = N(0,R)
where
% A - state transition matrix
% C - observation (output) matrix
% Q - state noise covariance
% R - observation noise covariance
% x0 - initial state mean
% P0 - initial state covariance
The code for the prediction and update are:
function [xpred, Ppred] = predict(x, P, F, Q)
xpred = A*x;
Ppred = A*P*A’ + Q;
function [nu, S] = innovation(xpred, Ppred, z, H, R)
nu = y - C*xpred; %% innovation
S = R + C*Ppred*C’; %% innovation covariance
function [xnew, Pnew] = innovation_update(xpred, Ppred, n
u, S, C)
K = Ppred*C’*inv(S); %% Kalman gain
xnew = xpred + K*nu; %% new state
Pnew = Ppred - K*S*K’; %% new covariance
I am trying to follow the implementation https://github.com/cswetenham/pmr/blob/master/toolboxes/lds/kalmansmooth.m but getting confused since the parameters A,C are given as matrices.
Do I have to represent A,C as matrices? How do I represent A,C , Q,R for my case?
How do I initialize A,C,Q,R? Since, I have only one variable, Q,R should be 1 by 1. But, what about the others?
Please help.
UPDATE: Here is the full implementation
%%% Matlab script to simulate data and process usiung Kalman for the state
%%% estimation of AR(2) time series.
% Linear system representation
% x_n+1 = A x_n + Bw_n
% y_n = Cx_n + v_n
% w = N(0,Q); v = N(0,R)
clc
clear all
T = 100; % number of data samples
order = 2;
% True coefficients of AR model
a1 = 0.195;
a2 = -0.95;
A = [ a1 a2;
0 1 ];
C = [ 1 0 ];
B = [1;
0];
x =[ rand(order,1) zeros(order,T-1)];
sigma_2_w =1; % variance of the excitation signal for driving the AR model(process noise)
sigma_2_v = 0.01; % variance of measure noise
Q=eye(order);
P=Q;
%Simulate AR model time series, x;
sqrtW=sqrtm(sigma_2_w);
%simulation of the system
for t = 1:T-1
x(:,t+1) = A*x(:,t) + B*sqrtW*randn(1,1);
end
%noisy observation
y = C*x + sqrt(sigma_2_v)*randn(1,T);
R=sigma_2_v*diag(diag(x));
R = diag(R);
z = zeros(1,length(y));
z = y;
x0=mean(y);
for i=1:T-1
[xpred, Ppred] = predict(x0,P, A, Q);
[nu, S] = innovation(xpred, Ppred, z(i), C, R);
[xnew, P] = innovation_update(xpred, Ppred, nu, S, C);
end
%plot
xhat = xnew';
plot(xhat(:,1),'red');
hold on;
plot(x(:,1));
function [xpred, Ppred] = predict(x0,P, A, Q)
xpred = A*x0;
Ppred = A*P*A' + Q;
end
function [nu, S] = innovation(xpred, Ppred, y, C, R)
nu = y - C*xpred; %% innovation
S = R + C*Ppred*C'; %% innovation covariance
end
function [xnew, Pnew] = innovation_update(xpred, Ppred, nu, S, C)
K = Ppred*C'*inv(S); %% Kalman gain
xnew = xpred + K*nu; %% new state
Pnew = Ppred - Ppred*K*C; %% new covariance
end
I have 3 problems:
(1) Is the initialization of the matrix R correct in terms of dimension and values?
R=sigma_2_v*diag(diag(x));
R = diag(R);
where x is the clean time series.
(2) When creating the noisy observation of the time series, I have used a different variance other than R. I have used sigma_2_v. Should I be using R or sigma_2_v? This is the line where I have created noisy time series
y = C*x + sqrt(sigma_2_v)*randn(1,T);
(3) The code is throwing the error
Error using inv
Matrix must be square.
Error in innovation_update (line 2)
K = Ppred*C'*inv(S); %% Kalman gain
Error in Kalman_run (line 65)
[xnew, P] = innovation_update(xpred, Ppred, nu, S, C);

Answer 1

There are different ways to represent a given AR process as a state space model but I believe you'll have to have a 2 dimensional state representation if it's an AR(2) process (which your model is). So A, C and Q will need to be matrices.
You have scalar observations and so C will be a 1x2 matrix.
x should be a 2 dimensional vector of the form [x(t); x(t-1)]. The first element is the value of x at time t from the AR model and the second element is x at time t-1.
A is of the form [a_1 a_2;1 0]. You can see that Ax is [a_1x(t)+a_2x(t-1);x(t)]. So after applying A to the state x the first element is the predicted next value of x (from the AR model a_1x(t)+a_2x(t-1)) and the second element is x at time t i.e. the elements have been shifted down.
C can be [1 0] i.e. the observation y at time t is simply the first element of the state vector x. Multiplying the state x by C should give use the observation of the state, here we want simply want to observe what the state is at time t and doing [1 0][x(t);x(t-1)] is just x(t).
A, Q, C and R can be learnt from data (e.g. as described here) or set using prior knowledge of your system.
Just because you have one observed variable at each time step doesn't mean Q and R are 1. Q is the system noise covariance and R is the observation noise covariance - setting them to one just specifies the covariance and doesn't do what you seem to think it does.